a farmer has 9500 meters of fencing, and wants to enclose a rectangular plot that borders on a river. if the…

Explanation:

Step1: Define the area function

Let the two equal - length sides perpendicular to the river be $x$ meters each, and the side parallel to the river be $y$ meters. The total length of the fence is $y + 2x=9500$, so $y = 9500 - 2x$. The area $A$ of the rectangle is $A(x)=x\cdot y=x(9500 - 2x)=9500x-2x^{2}$.

Step2: Find the derivative of the area function

Differentiate $A(x)$ with respect to $x$. Using the power rule, if $A(x)=9500x - 2x^{2}$, then $A^\prime(x)=\frac{d}{dx}(9500x)-\frac{d}{dx}(2x^{2})=9500 - 4x$.

Step3: Find the critical points

Set $A^\prime(x) = 0$ to find the critical points. So, $9500 - 4x=0$. Solving for $x$ gives $4x = 9500$, and $x=\frac{9500}{4}=2375$.

Step4: Determine the nature of the critical point

Differentiate $A^\prime(x)$ to get the second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(9500 - 4x)=-4<0$. Since $A^{\prime\prime}(x)<0$ when $x = 2375$, the function $A(x)$ has a maximum at $x = 2375$.

Step5: Calculate the maximum area

Substitute $x = 2375$ into the area function $A(x)$. $A(2375)=2375\times(9500 - 2\times2375)=2375\times(9500 - 4750)=2375\times4750 = 11281250$ square meters.

Answer:

$11281250$ square meters